Preheat the oven to 350 degrees F (175 degrees C). Grease a 12-cup muffin tin or line cups with paper liners.
Combine applesauce, oil, orange juice, egg, and vanilla extract in a large bowl; mix well.
Combine whole wheat flour, all-purpose flour, oats, brown sugar, white sugar, cinnamon, baking soda, and salt in another bowl. Add dry mix to wet mix, stirring just until moistened. Gently fold in cranberries; do not overmix. Divide batter evenly between the prepared muffin cups.
Bake in the preheated oven until tops spring back when lightly pressed and muffins are light golden brown, 15 to 25 minutes. Cool in the tin for 5 minutes, then transfer to a wire rack to cool completely.